Cedric Yen-Yu Lin

Hamiltonian simulation with optimal sample complexity

We investigate the sample complexity of Hamiltonian simulation: how many copies of an unknown quantum state are required to simulate a Hamiltonian encoded by the density matrix of that state? We show that the procedure proposed by Lloyd, Mohseni, and Rebentrost [Nat. Phys., 10(9):631–633, 2014] is optimal for this task. We further extend their method to the case of multiple input states, showing how to simulate any Hermitian polynomial of the states provided. As applications, we derive optimal algorithms for commutator simulation and orthogonality testing, and we give a protocol for creating a coherent superposition of pure states, when given sample access to those states. We also show that this sample-based Hamiltonian simulation can be used as the basis of a universal model of quantum computation that requires only partial swap operations and simple single-qubit states.Quantum Software from Quantum StatesOne of the hallmarks of quantum computation is the storage and extraction of information within quantum systems. Recently, Lloyd, Mohseni and Rebentrost created a protocol to treat multiple identical copies of a quantum state as “quantum software”, specifying a quantum program to be run on any other state. They use this approach to do principal component analysis of the software state. Here, we expand on their results, providing protocols for running more-complex quantum programs specified by several different states. Our protocols can be used to analyze the relationship between different states (for example, deciding whether states are orthogonal) and to create new states (such as coherent linear combinations of two states). We also outline the optimality of Lloyd et al.’s original protocol, as well as our new protocols.

Sequential measurements, disturbance and property testing

We describe two procedures which, given access to one copy of a quantum state and a sequence of two-outcome measurements, can distinguish between the case that at least one of the measurements accepts the state with high probability, and the case that all of the measurements have low probability of acceptance. The measurements cannot simply be tried in sequence, because early measurements may disturb the state being tested. One procedure is based on a variant of Marriott-Watrous amplification. The other procedure is based on the use of a test for this disturbance, which is applied with low probability. We find a number of applications: • Quantum query complexity separations in the property testing model for testing isomorphism of functions under group actions. We give quantum algorithms for testing isomorphism, linear isomorphism and affine isomorphism of boolean functions which use exponentially fewer queries than is possible classically, and a quantum algorithm for testing graph isomorphism which uses polynomially fewer queries than the best algorithm known. • Testing properties of quantum states and operations. We show that any finite property of quantum states can be tested using a number of copies of the state which is logarithmic in the size of the property, and give a test for genuine multipartite entanglement of states of n qubits that uses O(n) copies of the state. We also show that equivalence of two unitary operations under conjugation by a unitary picked from a fixed set can be tested efficiently. This is a natural quantum generalisation of testing isomorphism of boolean functions. • Correcting an error in a result of Aaronson on de-Merlinizing quantum protocols. This result claimed that, in any one-way quantum communication protocol where two parties are assisted by an all-powerful but untrusted third party, the third party can be removed with only a modest increase in the communication cost. We give a corrected proof of a key technical lemma required for Aaronsons result.

Space-Efficient Error Reduction for Unitary Quantum Computations

This paper develops general space-efficient methods for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness

Upper bounds on quantum query complexity inspired by the elitzur-vaidman bomb tester

c

Oracles with Costs

and soundness

Different Strategies for Optimization Using the Quantum Adiabatic Algorithm

s

Exponential Quantum Speed-ups for Semidefinite Programming with Applications to Quantum Learning.

, either with or without a witness (corresponding to QMA and BQP, respectively). To convert this computation into a new computation with error at most

Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems

2^{-p}

A Complete Characterization of Unitary Quantum Space.

, the most space-efficient method known requires extra workspace of

The computational power of normalizer circuits over black-box groups.

{O \bigl( p \log \frac{1}{c-s} \bigr)}